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Tree-based multivariate regression and density estimation with right-censored data

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  • Molinaro, Annette M.
  • Dudoit, Sandrine
  • van der Laan, M.J.Mark J.

Abstract

We propose a unified strategy for estimator construction, selection, and performance assessment in the presence of censoring. This approach is entirely driven by the choice of a loss function for the full (uncensored) data structure and can be stated in terms of the following three main steps. (1) First, define the parameter of interest as the minimizer of the expected loss, or risk, for a full data loss function chosen to represent the desired measure of performance. Map the full data loss function into an observed (censored) data loss function having the same expected value and leading to an efficient estimator of this risk. (2) Next, construct candidate estimators based on the loss function for the observed data. (3) Then, apply cross-validation to estimate risk based on the observed data loss function and to select an optimal estimator among the candidates. A number of common estimation procedures follow this approach in the full data situation, but depart from it when faced with the obstacle of evaluating the loss function for censored observations. Here, we argue that one can, and should, also adhere to this estimation road map in censored data situations. Tree-based methods, where the candidate estimators in Step 2 are generated by recursive binary partitioning of a suitably defined covariate space, provide a striking example of the chasm between estimation procedures for full data and censored data (e.g., regression trees as in CART for uncensored data and adaptations to censored data). Common approaches for regression trees bypass the risk estimation problem for censored outcomes by altering the node splitting and tree pruning criteria in manners that are specific to right-censored data. This article describes an application of our unified methodology to tree-based estimation with censored data. The approach encompasses univariate outcome prediction, multivariate outcome prediction, and density estimation, simply by defining a suitable loss function for each of these problems. The proposed method for tree-based estimation with censoring is evaluated using a simulation study and the analysis of CGH copy number and survival data from breast cancer patients.

Suggested Citation

  • Molinaro, Annette M. & Dudoit, Sandrine & van der Laan, M.J.Mark J., 2004. "Tree-based multivariate regression and density estimation with right-censored data," Journal of Multivariate Analysis, Elsevier, vol. 90(1), pages 154-177, July.
    • Handle: RePEc:eee:jmvana:v:90:y:2004:i:1:p:154-177
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    References listed on IDEAS

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    1. van der Laan Mark J. & Dudoit Sandrine & Keles Sunduz, 2004. "Asymptotic Optimality of Likelihood-Based Cross-Validation," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 3(1), pages 1-25, March.
    2. Sandra Sinisi & Mark van der Laan, 2004. "Loss-Based Cross-Validated Deletion/Substitution/Addition Algorithms in Estimation," U.C. Berkeley Division of Biostatistics Working Paper Series 1142, Berkeley Electronic Press.
    3. Keles Sunduz & van der Laan Mark J. & Dudoit Sandrine & Xing Biao & Eisen Michael B., 2003. "Supervised Detection of Regulatory Motifs in DNA Sequences," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 2(1), pages 1-40, August.
    4. Sandrine Dudoit & Mark van der Laan & Sunduz Keles & Annette Molinaro & Sandra Sinisi & Siew Leng Teng, 2004. "Loss-Based Estimation with Cross-Validation: Applications to Microarray Data Analysis and Motif Finding," U.C. Berkeley Division of Biostatistics Working Paper Series 1136, Berkeley Electronic Press.
    5. Leo Breiman & Jerome H. Friedman, 1997. "Predicting Multivariate Responses in Multiple Linear Regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(1), pages 3-54.
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    Cited by:

    1. Sinisi Sandra E. & Neugebauer Romain & van der Laan Mark J., 2006. "Cross-Validated Bagged Prediction of Survival," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 5(1), pages 1-26, May.
    2. Karen Lostritto & Robert L. Strawderman & Annette M. Molinaro, 2012. "A Partitioning Deletion/Substitution/Addition Algorithm for Creating Survival Risk Groups," Biometrics, The International Biometric Society, vol. 68(4), pages 1146-1156, December.
    3. Olivier Lopez & Xavier Milhaud & Pierre-Emmanuel Thérond, 2015. "Tree-based censored regression with applications to insurance," Working Papers hal-01141228, HAL.
    4. Laan Mark J. van der & Dudoit Sandrine & Vaart Aad W. van der, 2006. "The cross-validated adaptive epsilon-net estimator," Statistics & Risk Modeling, De Gruyter, vol. 24(3), pages 1-23, December.
    5. Yan Zhou & John McArdle, 2015. "Rationale and Applications of Survival Tree and Survival Ensemble Methods," Psychometrika, Springer;The Psychometric Society, vol. 80(3), pages 811-833, September.
    6. Mark van der Laan & Sandrine Dudoit & Aad van der Vaart, 2004. "The Cross-Validated Adaptive Epsilon-Net Estimator," U.C. Berkeley Division of Biostatistics Working Paper Series 1141, Berkeley Electronic Press.
    7. Wei-Yin Loh, 2014. "Fifty Years of Classification and Regression Trees," International Statistical Review, International Statistical Institute, vol. 82(3), pages 329-348, December.
    8. Olivier Lopez & Xavier Milhaud & Pierre-Emmanuel Thérond, 2016. "Tree-based censored regression with applications in insurance," Post-Print hal-01141228, HAL.
    9. Susan Athey & Julie Tibshirani & Stefan Wager, 2016. "Generalized Random Forests," Papers 1610.01271, arXiv.org, revised Apr 2018.
    10. Yifei Sun & Sy Han Chiou & Mei‐Cheng Wang, 2020. "ROC‐guided survival trees and ensembles," Biometrics, The International Biometric Society, vol. 76(4), pages 1177-1189, December.
    11. Alexander Hanbo Li & Jelena Bradic, 2019. "Censored Quantile Regression Forests," Papers 1902.03327, arXiv.org.
    12. Pablo Gonzalez Ginestet & Ales Kotalik & David M. Vock & Julian Wolfson & Erin E. Gabriel, 2021. "Stacked inverse probability of censoring weighted bagging: A case study in the InfCareHIV Register," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 70(1), pages 51-65, January.

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